Robust Optimization Approach Research Articles

Fabrication uncertainty guided design optimization of a photonic crystal cavity by using Gaussian processes

We present a fabrication uncertainty aware and robust design optimization approach that can be used to obtain robust design estimates for nonlinear, nonconvex, and expensive model functions. It is founded on Gaussian processes and a Monte Carlo sampling procedure, and assumes knowledge about the uncertainties associated with a manufacturing process. The approach itself is iterative. First, a large parameter domain is sampled in a coarse fashion. This coarse sampling is used primarily to determine smaller candidate regions to investigate in a second, more refined sampling pass. This finer step is used to obtain an estimate of the expected performance of the found design parameter under the assumed manufacturing uncertainties. We apply the presented approach to the robust optimization of the Purcell enhancement of a photonic crystal nanobeam cavity. We obtain a predicted robust Purcell enhancement of F¯P≈3.6. For comparison we also perform an optimization without robustness. We find that an unrobust optimum of FP≈256.5 dwindles to only F¯P≈0.7 when fabrication uncertainties are taken into account. We thus demonstrate that the presented approach is able to find designs of significantly higher performance than those obtained with conventional optimization.

Consignment contract in a supply chain with price, demand and yield randomness: A distributionally robust approach

We investigate a supply chain consisting of a supplier and an e-commence platform who sign a consignment contract with uncertainty in price, demand, and yield. Due to the challenges of actually determining the joint distribution of random variables, we assume that only the partial distributional information is available for these three uncertainties. To address the issue of distributional ambiguity and to more accurately account for arbitrary correlations among these uncertainties, we introduce a moment-based distributionally robust optimization (DRO) approach that assumes only the first and second moments of these three uncertainties are known. Given an exogenous consignment rate, we derive the closed-form expression for the supplier’s optimal production decision and her worst-case expected profit. We study the impact of random price and random yield within the ambiguity set on the supplier’s production decision and the worst-case expected profit. From the e-commerce platform’s perspective, we further examine the contract design problem when the consignment rate is endogenous and analyze equilibrium solutions for both contracting parties. Our results show that the additional consideration of price or yield uncertainty within the ambiguity set does not always mitigate or enhance the degree of conservatism of the contracting parties, which is closely related to the correlation between any two types of uncertainty, as well as the degree of uncertainty itself.

A robust optimization approach to multi-period competitive location problem for bank branches considering first-mover advantage

This paper proposes a robust, dynamic, multi-objective optimization approach to the capacity-location problem for bank branches in a competitive environment named Multi-Period Competitive Location (MPCLP) Problem. The objective functions include, maximizing total profit and maximizing minimum distance between active branches. The decision variables include branch merger and acquisition (M&A), branch establishment and determination of branch capacities at each period. Market share is calculated using a modified Huff gravity function that considers distance and design (capacity) of the branch, as well as the effect of first-mover advantage. In other words, in this model, branches established sooner have comparative advantage over new comers. The resultant bi-objective model is transformed into a single objective nonlinear mixed integer programming problem using Epsilon Constraint technique. To deal with uncertainty in demand, a scenario-based robust optimization approach is applied. Given the NP-hard nature of the proposed model, we apply a hybrid Genetic Algorithm-Variable Neighbourhood Search (GA-VNS) algorithm to deal with large-scale problems. We demonstrate the performance of our proposed algorithm on a real case study in northern Tehran and validate the results against those obtained from exact algorithms in small-sized problems, as well as single genetic and variable neighbourhood search algorithms in large-scale problems. Our results show that our algorithm's performance in small-scale problems is comparable to that provided by exact algorithms, while also outperforming both genetic algorithm and variable neighbourhood search in most large-scale problems.

A novel implicit decision variable classification approach for high-dimensional robust multi-objective optimization in order scheduling

This paper efficiently addresses the high-dimensional robust order scheduling problem. A novel algorithm named dynamic cooperative coevolution based on an implicit decision variable classification approach (DCC/IDVCA) is developed to search for robust order schedules. To significantly reduce the computational resources required for solving the high-dimensional robust order scheduling problem, we propose decomposing the original decision variables through implicit classification methods. First, a novel estimation method is introduced to evaluate the weighted contribution of variables to robustness. This method utilizes historical information, including the variation of the overall mean effective fitness and the frequency of variables being classified into highly robustness-related subcomponents in previous cycles, for evaluating their weighted contribution to robustness. Then, based on the corresponding weighted robustness contributions, the original variables are classified into highly and weakly robustness-related variables. Finally, these two types of variables are decomposed into highly and weakly robustness-related subgroups within a dynamic cooperative coevolution framework and optimized separately. In the experimental section, the proposed algorithm is applied to two practical order scheduling problems in discrete manufacturing industry. The experimental results demonstrate that the proposed algorithm achieves competitive outcomes compared to state-of-the-art high-dimensional robust multi-objective optimization algorithms.

A sample robust optimal bidding model for a virtual power plant

In many energy markets, the trade amount of electricity must be committed to before the actual supply. This study explores one consecutive operational challenge for a virtual power plant—the optimal bidding for highly uncertain distributed energy resources in a day-ahead electricity market. The optimal bidding problem is formulated as a scenario-based multi-stage stochastic optimization model. However, the scenario-tree approach raises two consequent issues—scenario overfitting and massive computation cost. This study addresses the issues by deploying a sample robust optimization approach with linear decision rules. A tractable robust counterpart is derived from the model where the uncertainty appears in a nonlinear objective and constraints. By applying the decision rules to the balancing policy, the original model can be reduced to a two-stage stochastic mixed-integer programming model and then efficiently solved by adopting a dual decomposition method combined with heuristics. Based on real-world business data, a numerical experiment is conducted with several benchmark models. The results verify the superior performance of our proposed approach based on increased out-of-sample profits and decreased overestimation of in-sample profits.

Multi-Objective Robust Trajectory Design for Powered Descent and Landing

A robust trajectory optimization approach for guidance algorithm gain and target vector selection for powered descent and landing is developed. A genetic algorithm is used to determine optimized guidance algorithm parameters to minimize the impact of initial condition, environment, navigation, and vehicle property uncertainty on flight performance for a given sensor suite. Vehicle state uncertainties are computed efficiently using linear covariance analysis techniques. When implemented in the guidance algorithm, the optimized gains and target vectors shape a trajectory that has more favorable conditions for a given navigation sensor suite, resulting in improved flight performance. As a demonstration of this method, the optimized guidance parameters are found for a multiphase trajectory from powered descent initiation to touchdown for a robotic lunar landing mission. Single-objective optimization results demonstrate a reduction in uncertainty and an improvement in nominal performance. Multi-objective optimization results showing the tradeoff between terminal position uncertainty and total propellant usage are presented for multiple sensor suite compositions. Further, guidance parameters selected using the developed robust trajectory design approach may enable acceptable flight performance with fewer and/or lower-quality sensors. The resulting Pareto fronts present the optimal trade space early in the mission design process to enable informed decision-making.

Multi-Objective Robust Optimization of Integrated Energy System with Hydrogen Energy Storage

A novel multi-objective robust optimization model of an integrated energy system with hydrogen storage (HIES) considering source–load uncertainty is proposed to promote the low-carbon economy operation of the integrated energy system of a park. Firstly, the lowest total system cost and carbon emissions are selected as the multi-objective optimization functions. The Pareto front solution set of the objective function is applied by compromise planning, and the optimal solution among them is obtained by the maximum–minimum fuzzy method. Furthermore, the robust optimization (RO) approach is introduced to cope with the source–load uncertainty effectively. Finally, it is demonstrated that the illustrated HIES can significantly reduce the total system cost, carbon emissions, and abandoned wind and solar power. Meanwhile, the effectiveness of the proposed model and solution method is verified by analyzing the influence of multi-objective solutions and a robust coefficient on the Chongli Demonstration Project in Hebei Province.

Data-driven distributionally robust optimization approach for the coordinated dispatching of the power system considering the correlation of wind power

With the increasing penetration of large-scale wind power into the power grid, it is crucial to develop a precise model to accurately depict the stochasticity and correlation among wind farm outputs, which is highly important for ensuring the safe and efficient utilization of wind energy in grid dispatching. In this article, a data-driven distributionally robust optimization (DDRO) dispatching approach that accounts for spatial correlations among outputs from multiple wind farms is proposed. The proposed approach is applied to a source-network-load-storage grid system to ascertain unit start-stop schedules and resource allocation effectively. First, a truncated spatial correlation model is proposed, enabling a comprehensive representation of spatial correlations and output constraints between distinct wind farms. Second, the ISODATA clustering algorithm is employed to generate typical scenarios, reduce model complexity, and expedite the computation process. Third, a unit commitment model considering the demand response is constructed and solved using the DDRO approach. Finally, the proposed model is applied to the IEEE 30-bus system to test its robustness and cost-effectiveness compared to the traditional robust optimization model. Additionally, it is applied to the IEEE 118-bus system to demonstrate its scalability and stability.

Optimal energy storage system design for addressing uncertainty issues in integration of supply and demand-side management approaches

The primary goal of Sustainable Development Goal 7 (SDG 7) is to increase renewable energy use to reduce reliance on fossil fuels and mitigate climate change. Energy-intensive industries can benefit from in-house renewable power generation, reducing their reliance on fossil fuel-based grid power and making processes greener. However, integration among power generation/purchase, energy storage systems (ESS), and power consumption is crucial to overcome the intermittent nature of renewable power sources. ESS plays a vital role in increasing resilience and optimizing material production based on power availability and pricing. The efficacy of ESS needs critical evaluation considering cost, efficiency, and uncertainties related to renewable power generation and material product demand. The paper proposes two scenario-based optimization approaches to assess the impact of uncertainties on the integrated supply and demand side management (ISDM) system, focusing on lithium-ion batteries and cryogenic energy storage (CES). Compared to the conservative approach of the scenario-based robust optimization (SRO) method, the proposed stochastic simulation optimization (SSO) method provides a ‘risk-neutral’ solution, which is 6.45% less than the minimum expected cost solution. The analysis also suggests that lithium-ion batteries are more economically effective for the proposed integrated framework than CES, resulting in almost a 29% reduction in operating costs compared to no battery option. The proposed framework could contribute to sustainable and economically viable energy management practices in energy-intensive industries. Further research and implementation of such frameworks could accelerate the adoption of renewable energy and energy storage technologies in industrial processes.

Sustainable supply chain planning with the flexible provisions of inter-period emission credits borrowing and banking under a multi-period carbon trading scheme

Although the carbon emission trading regulation proves its advantage of reducing abatement costs, the timing of operating activities makes overage or shortage of allocated emission allowances quickly happen in single-year compliance, thus increasing abatement costs. The provision of inter-period emission credits banking and borrowing in a multi-period compliance phase can give emitters flexibility in emission compliance, which practices world-widely. In this study, we investigate the impact of carbon trading with inter-temporal emission credits borrowing and banking on a production–inventory–distribution system’s supply chain network decision-making and its compliance strategies by proposing a novel two-stage stochastic programming. A scenario-based robust optimization approach is presented with a post-optimization stage to obtain robustness-profit-environment balanced decision-making considering uncertain demand and stochastic carbon-trading price. The results suggests emitters should adopt specific greener techniques to reduce related emissions with priority under carbon-trading policies. The results also show the provision of carbon credits borrowing and banking has a positive influence on enhancing the efficiency in supply chain economic and ecological performance. Besides, given the flexible provisions, the manager can reallocate emission credits according to its business’s operation, and transfer more emission credits to periods of higher carbon-trading price. This findings also suggest banking in early periods instead of carelessly borrowing from future, for uncertainty in future business might make it harder or impossible to comply with emission regulation at last. To regulators, it should prevent borrowing explicitly or allow it only to a limited extent.

Data-driven-based distributionally robust optimization approach for a virtual power plant considering the responsiveness of electric vehicles and Ladder-type carbon trading

To encourage the utilization of decentralized renewable energy systems, a data-driven-based distributionally robust optimization (DRO) model is proposed for a virtual power plant (VPP) considering the responsiveness of electric vehicles (EVs) and a ladder-type carbon trading mechanism (LT-CTM). An EV virtual energy storage (EV-VES) calculation model for dispatchable power is formulated to facilitate the centralized scheduling of EVs, considering responsiveness and spatial–temporal distribution characteristics. Additionally, multiple kinds of energy storage systems and flexible loads are considered in the VPP. Furthermore, to handle multiple uncertainties for both the renewable energy and load, Latin hypercube sampling is applied to generate the renewable energy scenarios, and K-means clustering is used to obtain typical scenes. Then the DRO model is used to solve the dispatch problem. An LT-CTM is leveraged to reduce carbon emissions. The column and constraint generation algorithm is then employed to identify an optimal operational benefit solution. Finally, a multi-parameter coupling analysis of ladder-type carbon trading is used for the environmental protection and economy of the VPP. The results confirm the efficacy of the proposed day-ahead scheduling model, offering a theoretical reference for operators when making scheduling decisions.

A robust optimization approach for placement of applications in edge computing considering latency uncertainty

Edge computing brings computing and storage resources close to end-users to support new applications and services that require low network latency. It is currently used in a wide range of industries, from industrial automation and augmented reality, to smart cities and connected vehicles, where low latency, data privacy, and real-time processing are critical requirements. The latency of accessing applications in edge computing must be consistently below a threshold of a few tens of milliseconds to maintain an acceptable experience for end-users. However, the latency between users and applications can vary considerably depending on the network load and mode of wireless access. An application provider must be able to guarantee that requests are served in a timely manner by their application instances hosted in the edge despite such latency variations. This article focuses on the placement and traffic allocation problem faced by application providers in determining where to place application instances on edge nodes such that requests are served within a certain deadline. It proposes novel formulations based on robust optimization to provide optimal plans that protect against latency variations in a configurable number of network links. The robust formulations are based on two different types of polyhedral uncertainty sets that offer different levels of protection against variations in latency. Extensive simulations show that our robust models are able to keep the number of chosen edge nodes low while reducing the number of latency violations as compared to a deterministic optimization model that only considers the average latency of network links.

A robust optimization approach for repairing and overhauling in a captive repair shop under uncertainty

A robust optimization approach for repairing and overhauling in a captive repair shop under uncertainty

A robust optimization approach for resiliency improvement in power distribution system

AbstractThe occurrence of natural disasters has led to an alarming increase in power interruptions, with severe impacts. Compounding this problem is the uncertain nature of data, which presents significant challenges in enhancing the resiliency of power distribution systems following such events. To tackle these issues, this paper introduces a robust optimization approach for improving the resiliency of power distribution systems. The approach encompasses crew teams for switching actions as part of the restoration process, along with demand response programs and mobile generators (MGs). By simultaneously leveraging these elements and considering the uncertainty associated with electrical load and electrical price, the resiliency of the system is enhanced. The objective function is tri‐level, comprising minimum, maximum, and minimum functions. At the first level, the approach minimizes the cost of commitment of combined heat and power plants (CHPs) by taking into account the location of MGs and the reconfiguration structure in power distribution systems. The second level aims to identify the worst‐case scenario for the uncertainty variables. Finally, the third level focuses on minimizing the total operation cost under the worst‐case scenario using demand response programs. The proposed algorithm is implemented on an IEEE 33‐bus test distribution system, with four different cases investigated.

Design Optimisation Approach of an Outer Rotor Multiphase PM Actuator for Multirotor Aerial Vehicle Applications

The electric urban air mobility sector has gained significant attraction in public debates, particularly with the proliferation of announcements demonstrating new aerial vehicles and the infrastructure that goes with them. In this context, the development of new methodologies for the design and sizing of actuation systems, ensuring high performances of these aerial vehicles, remains an important task in this process. This will allow for better integration within this transport sector. In this paper, a robust design optimisation approach of multiphase fault-tolerant (FT) outer rotor (OR) permanent magnets (PM) for multirotor aerial vehicle applications is proposed. In order to show the effectiveness and the robustness of the proposed design methodology, the number of stator winding phases, with a fractional slot concentrated winding (FSCW) configuration, as well as the PM configuration are considered as variables. Thus, four cases for the number of phases are considered, namely 3, 5, 6 and 7 phases, where for each number of phases case, the PM takes 3 configurations, namely surface PM, interior V-shape PM and interior spoke PM. First, a pre-sizing step is carried out, consisting of selecting the optimal combinations slot/pole, designing the multiphase FSCW layout, and estimating the electric motor (EM) geometry using analytical computations to obtain a preliminary validation of the design specifications. Second, constrained multiobjective optimisation is considered in order to optimise the EM performances, such as motor efficiency and weight, under constraints where the FEMM/Matlab based Finite Element Analysis (FEA) tool is used to perform this optimisation. Finally, results analysis and performance comparisons of different EM configurations are carried out in order to assess the design parameters, such as phases number, PM position, and harmonic currents in the EM design and consequently to select the best configuration for the considered application.

Development of a novel multi-fidelity meta modeling approach for robust multi-objective optimization of a natural gas-hydrogen/diesel dual fuel engine

Multi-fidelity modeling (MFM) is an evolving field that matches low-fidelity models (LFM) and high-fidelity models (HFM) to get better solutions with low computational cost. However, improving the duality between accuracy and computational cost remain challenging, particularly for complex problems such as dual fuel engines. This paper contributes to the MF modeling cost-effectiveness improvement by proposing a new approach to solve large-dimensional multi-objective optimization problems. The first step is to build a meta-model based on the LF model, which will be subjected to a comet-governed analysis to detect potential areas where the uncertainty on the LF model is relatively high. Then, a design of experiment (DOE) will be developed based on the results of this analysis to construct an initial HF model. Finally, an iterative loop will be activated to improve the accuracy of the MF model using a well-weighed combination of the details delivered by the LF model correction via the HF model and the HF meta-model. The developed approach is validated on four different mathematical benchmarks with different difficulties, compared with four different MF modeling strategies. This validation shows that the proposed MF modeling is competitive and can produce solutions as accurate as the HF model while reducing significantly the overall computation time by up to 50%. As an engineering application, the operating conditions in a natural gas-hydrogen/diesel dual fuel engine in terms of compression ratio, pilot injection timing, and EGR are optimized. A reduction of 46%, 68%, and 96% was achieved for HC, NOx, and the knocking index, respectively, while an increase in thermal efficiency of about 5% was obtained.

Robust Collaborative Scheduling Strategy for Multi-Microgrids of Renewable Energy Based on a Non-Cooperative Game and Profit Allocation Mechanism

The Multiple Microgrid System (MMG) facilitates synergistic complementarity among various energy sources, reduces carbon emissions, and promotes the integration of renewable energy generation. In this context, we propose a two-stage robust cooperative scheduling model for MMGs based on non-cooperative game theory and a benefit allocation mechanism. In the first stage, considering electricity price fluctuations and uncertainties in wind and solar power outputs, a robust optimization approach is applied to establish an electric energy management model for MMGs. This model enables point-to-point energy sharing among microgrids. In the second stage, addressing the benefit allocation problem for shared electric energy, we introduce a Cost Reduction Ratio Distribution (CRRD) model based on non-cooperative game theory. The generalized Nash equilibrium is utilized to determine the benefit distribution for shared electric energy. Finally, through case studies, the proposed model is validated, ensuring fair returns for each microgrid. The results indicate that the proposed model optimizes the operational states of individual microgrids, reduces operational costs for each microgrid, and lowers the overall total operational costs of the MMG system. Additionally, an investigation is conducted into the impact of electricity price uncertainty coefficients and confidence levels of wind and solar uncertainties on the operational costs of microgrids.

Dynamic optimization of blood collection strategies from different potential donors using rolling horizon planning approach under uncertainty

Dynamic blood collection planning is crucial for adjusting strategies in response to varied donors' behavior within the dynamic urban environment, which can be utilized to mitigate wastage and scarcity due to the uncertain supply chain parameters. In this study, we address the challenge of adapting collection planning to dynamic fluctuations in potential blood donors’ behavior by introducing an innovative updatable approach based on the rolling horizon planning approach, which considers uncertain conditions in daily urban life. Taking into account one-day planning and diverse urban areas contexts, we propose an approach that operationally updates relocation and collection strategies throughout the daily fluctuation in urban conditions. We present a robust optimization approach to mitigate uncertainty across various parameters such as donors' behavior, demand, transportation time, and operational cost. Given the impact of dynamic and diverse conditions, we implement our updatable approach within the Tehran blood supply chain, which possesses the largest blood network in northern Iran, providing support to 108 hospitals and numerous laboratories. Deriving insights from our results, we prove a noteworthy 9.65% enhancement in blood collection across the entire network resulting from the implementation of the rolling horizon planning approach. Moreover, adjusting the dynamic conditions, in contrast to non-dynamic conditions, has led to an additional 24.14% increase in blood collection through temporary collection facilities as a main focus in this investigation.

Multi-objective programming for multi-period multi-product closed-loop supply chain network design: a fuzzy robust optimization approach

Multi-objective programming for multi-period multi-product closed-loop supply chain network design: a fuzzy robust optimization approach

Linear quadratic mean-field game with volatility uncertainty

In this paper, we consider a linear quadratic (LQ) mean field game (MFG) with uncertain volatility. Applying a robust optimization approach and based on mean-field approximation, we firstly obtain the worst uncertainty by dealing with an LQ MFG with so-called soft-constraint. Then with the help of convex analysis theory, we derive the auxiliary optimal control problems for the agents and construct the decentralized strategies. Finally, by virtue of estimates of (forward) backward stochastic differential equations, the asymptotical optimality of the decentralized strategy is verified. A numerical example is given to illustrate the above theoretical results.